Quantifying the localization uncertainty of modulation enhanced single-molecule localization microscopy

Abstract

To assist in medicine development and microbiological research, microscopy has been an important tool ever since the seventeenth century. Fluorescence microscopy is able to provide the specificity and contrast needed for biological imaging, and the physical resolution limit caused by diffraction can be circumvented through super-resolution microscopy. By combining modulated excitation with sparse activation of fluorescent emitters and subsequent localization of emitter positions, modulation enhanced single molecule localization microscopy achieves a localization precision in the order of magnitude of nanometres to Angstroms, thereby making the invisible visible.

While these super-resolution methods allow us to access the nanoscale, their findings are accompanied by the statistical uncertainty about whether the molecule positions that we retrieve correspond to the true underlying positions of emitters that are truly present in the sample. A fundamental objective of super-resolution microscopy is thus to give certainty about the localization uncertainty with which the position of a single molecule can be determined. To make the uncertain certain in single-molecule localization, the Cramer-Rao lower bound is commonly used. The Cramer-Rao lower bound represents the theoretical minimum uncertainty with which unbiased estimators can localize emitters. However, the Cramer-Rao lower bound leads to narrowly applicable, improperly represented or mathematically incorrect characterizations of the localization precision of modulation enhanced single-molecule localization microscopy.

To address this, new and generalizable image formation models are needed. In addition, we need to develop statistical tools that represent the full estimator distribution, as well as the uncertainty of localization methods that use biased estimators.

In this dissertation, we address these issues through three major contributions. As our first contribution, we derive a new and generalizable image formation model that integrates modulation enhanced localization in existing setups that use a spinning disk in the illumination- and emission paths, leading to the theoretical design of a new method called SpinFlux. In the SpinFlux analysis, emitters are localized in the recordings from a sequence of individual pattern acquisitions, taking knowledge about the pattern into account. SpinFlux shows its merit when the excitation intensity is modulated to incorporate the maximum amount of information, reaching a 3.5-fold local precision improvement over single-molecule localization microscopy when using donut-shaped illumination patterns. Combined with the versatility of the image formation model to incorporate arbitrary illumination patterns, this makes SpinFlux the method of choice for local refinements of the localization precision.

Secondly, we analyse the occurrence of multimodality in three-dimensional multiple emitter imaging by reconstructing the full posterior distribution of localization. We develop a Bayesian three-dimensional localization method called three-dimensional reversible jump Markov chain Monte Carlo, which approximates the posterior density of emitter positions rather than giving point estimates. We show that astigmatic multiple emitter imaging results in a multimodal posterior distribution when two emitters are separated by less than the standard deviation of the in-focus point spread function, which causes ambiguous solutions to the estimation problem. This motivates the importance of including appropriately chosen uncertainty measures in localization algorithms. In particular, estimation of the full posterior distribution makes it possible to detect cases where the localization uncertainty for individual emitters is not accurately represented by Gaussian uncertainty ellipses, which would be misrepresented by the Cramer-Rao lower bound.

Lastly, we quantify and analyse the localization precision of iterative localization microscopy methods, such as MINFLUX. These methods are able to locally improve the localization precision around an emitter position by using prior information derived from measurements in earlier iterations. As the Cramer-Rao lower bound requires estimators to be unbiased, it cannot incorporate prior information, making it inapplicable to iterative localization microscopy. However, the Bayesian Van Trees inequality circumvents this mathematical limitation, and is therefore an appropriate bound to analyse iterative localization microscopy. By taking modulation- and background imperfections into account, we show that the improvement of iterative methods over single-molecule localization is at most fivefold. The Van Trees inequality allows us to nuance existing precision limits for methods resembling MINFLUX when affected by modulation- and background imperfections, by showing that the precision of these methods is not maximized by minimizing the pattern distance, nor exponentially improved by increasing the iteration count.

Based on these findings we argue that, in order to reflect the statistical uncertainty of the localization process, emitter position estimates in single-molecule localization microscopy should be presented in the context of the estimation uncertainty. Image formation models and uncertainty quantification should be tailored to the application, letting the particularities of the application determine the choice of appropriate mathematical tools. As shown in this dissertation, this attitude towards uncertainty leads to new experimental methods to improve the localization precision, and it advances our fundamental understanding of localization uncertainty in super-resolution microscopy.